\contentsline {section}{\numberline {1}Integration on $({\cal H}_p\times {\cal H})/\Gamma $}{11}
\contentsline {subsection}{\numberline {1.1}Forms on ${\cal T}\times {\cal H}$}{11}
\contentsline {subsection}{\numberline {1.2}Mixed period integrals}{19}
\contentsline {section}{\numberline {2}Periods attached to split tori}{25}
\contentsline {subsection}{\numberline {2.1}Preliminary calculations}{25}
\contentsline {subsection}{\numberline {2.2}Evaluation of $\unhbox \voidb@x \hbox {ord}_p(I_\Psi )$}{31}
\contentsline {subsection}{\numberline {2.3}Evaluation of $\qopname \relax o{log}(I_\Psi )$}{36}
\contentsline {section}{\numberline {3}The cohomology of $\Gamma $}{38}
\contentsline {subsection}{\numberline {3.1}The cohomology of $M$-symbols}{38}
\contentsline {subsection}{\numberline {3.2}Proof of theorem 4\hbox {}}{42}
\contentsline {subsection}{\numberline {3.3}Indefinite integrals}{43}
\contentsline {section}{\numberline {4}Periods attached to real quadratic fields}{45}
\contentsline {section}{\numberline {5}Heegner points attached to real quadratic fields}{47}
\contentsline {subsection}{\numberline {5.1}The main conjecture}{47}
\contentsline {subsection}{\numberline {5.2}A Shimura Reciprocity Law}{50}
\contentsline {subsection}{\numberline {5.3}A Gross-Zagier conjecture}{55}
\contentsline {subsection}{\numberline {5.4}Numerical evidence }{55}